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Questions Related to maths

If $p$ is any statement $t$ and $c$ are tautology and contradiction respectively, then which of the following is(are) correct?

logical equivalence mathematical logic discrete mathematics business maths maths

  1. $p\wedge t \equiv p$

  2. $p \wedge c \equiv c$

  3. $p\vee t \equiv c$

  4. $p \vee c \equiv p$

Show answer
Correct Option: A,B,D
Explanation:

Tautology is the preposition which is always true and contradiction is a preposition which is always false.
Here $\wedge \equiv AND, \vee \equiv OR$, $T$=true and $F$=false
Given statement=$p$, tautology=$t$ and contradiction=$c$
a)$p\wedge t\equiv p$ irrespective of value of $p$ as $t=T$ always
And if $p=T$ then $T\wedge T=T$ and if $p=F$ then $F\wedge T=F$
Thus option (a) is correct.
b)$p\wedge c\equiv c$ irrespective of value of $p$ as$ c=F$ always
And if $p=T$ then $T\wedge F=F$ and if $p=F$ then $F\wedge F=F$
Thus option (b) is correct
c)$p\vee t\equiv c$
If $p=T$ then $T\vee T\equiv T\neq c$ and if $p=F$ then $F\vee T\equiv T \neq c$
Thus option (c) is not correct
d)$p\vee c\equiv p$
If $p=T$ then $T\vee F\equiv T$ and if $p=F$ then $F\vee F\equiv F$
Thus it depends on the value of $p$
Hence option (d) is correct

Which one of the following statements is a tautology?

logical equivalence mathematical logic discrete mathematics business maths maths

  1. $\left( p\vee q \right) \rightarrow q$

  2. $p\vee (p\rightarrow q)$

  3. $ p\vee (q\rightarrow p)$

  4. $p\rightarrow (p\rightarrow q)$

Show answer
Correct Option: B
Explanation:
$ p$  $q$  $p\to q$  $q\to p$ $pvq$  $p\to (p\to q)$  $(pvq)\to q$  $pv(p\to q)$   $pv(q\to q$
 T  T  T  T  T  T  T  T  T
 T  F  F  T  T  F  F  T  T
 F  T  T  F  T  T  T  T  F
 F  F  T  T  F  T  T  T  T

$\left( p\vee q \right) \rightarrow q$

If $p$ and $q$ are two statement, then $(p  \wedge \sim q) \wedge   (\sim p  \wedge   q)$ is

logical equivalence mathematical logic discrete mathematics business maths maths

  1. a fallacy

  2. a tautology

  3. neither tautology nor a fallacy

  4. none of these

Show answer
Correct Option: A
Explanation:
$p$         $q$          $p \land \sim q$       $q \land \sim p $          $(p \land \sim q ) \land(q \land \sim p )$
T T     F       F          F
T F     T       F          F
F T      F       T          F
F F     F       F          F


$\therefore\ (p \land \sim q ) \land(q \land \sim p )$ is a fallacy

Which of the following is equal to  $1?$

maths calculations and mental strategies 4 mental additions and subtractions of decimals multiplication and division of decimals mental multiplication and division

  1. $\dfrac { 0.304 \times 20 } { 304 \times 2 }$

  2. $\dfrac { 0.304 \times 20 } { 3 \cdot 04 \times 2 }$

  3. $\dfrac { 0.304 \times 2 } { 30 \cdot 4 \times 2 }$

  4. $\dfrac { 0.304 \times 2 } { 304 \times 0 \cdot 2 }$

Show answer
Correct Option: B
Explanation:

$\begin{array}{l} \dfrac { { 0.304\times 20 } }{ { 3.04\times 2 } }  \ =\dfrac { { 304\times 20\times 100 } }{ { 304\times 2\times 1000 } }  \ =1 \ Hence, \ B\, is\, the\, correct\, answer. \end{array}$

What decimal of an hour is a second ?

maths calculations and mental strategies 4 mental additions and subtractions of decimals multiplication and division of decimals mental multiplication and division

  1. 0.0025

  2. 0.00027

  3. 0.0256

  4. 0.000126

Show answer
Correct Option: B
Explanation:

Required decimal = $\dfrac{1}{60 \times 60}=\dfrac{1}{3600}=0.00027$

If $4.175 = \displaystyle\frac { 1 }{ 0.2395 } $, then what is $\displaystyle\frac { 1 }{ 0.0004175 } $ equal to ?

maths calculations and mental strategies 4 mental additions and subtractions of decimals multiplication and division of decimals mental multiplication and division

  1. 0.2395

  2. 2395

  3. 2.395

  4. 23.95

Show answer
Correct Option: B
Explanation:

$\displaystyle\frac { 1 }{ 0.0004175 } = \displaystyle\frac { 1 }{ \displaystyle\frac { 4.175 }{ 10000 }  } = \displaystyle\frac { 10000 }{ 4.175 } = \displaystyle\frac { 10000 }{ \displaystyle\frac { 1 }{ 0.2395 }  } $

$= 10000 \times 0.2395 = 2395$

Evaluate : $\left( 78.34+96.68-14.44 \right) \div 4$.

maths calculations and mental strategies 4 mental additions and subtractions of decimals multiplication and division of decimals mental multiplication and division

  1. $34.145$

  2. $16.58$

  3. $40.145$

  4. $45.346$

Show answer
Correct Option: C
Explanation:

$\left( 78.34+96.68-14.44 \right) \div 4=160.58\div 4=40.145$

$\displaystyle \frac{\left ( 0.35 \right )^{2}-\left ( 0.03 \right )^{2}}{0.19}=? $

maths calculations and mental strategies 4 mental additions and subtractions of decimals multiplication and division of decimals mental multiplication and division

  1. 0.32

  2. 0.48

  3. 0.76

  4. 0.64

Show answer
Correct Option: D
Explanation:

$\dfrac{(0.35)^{2}-(0.03)^{2}}{0.19}$=$\dfrac{(0.35+0.03)(0.35-0.03)}{0.19}$    $[a^2-b^2=(a+b)(a-b)]$


$=\dfrac{(0.38)(0.32)}{0.19}$


$=2\times{0.32}$

$=0.64$

So the correct answer will option D

What is $\displaystyle 0.\overline{09}\times7.\overline{3}$ equal to ?

maths calculations and mental strategies 4 mental additions and subtractions of decimals multiplication and division of decimals mental multiplication and division

  1. $\displaystyle 0.\overline{6}$

  2. $\displaystyle 0.\overline{7}$

  3. 0.6

  4. 0.7

Show answer
Correct Option: A
Explanation:

$\displaystyle 0.\overline{09}\times7.\overline{3}=\frac{9}{99}\times7\frac{3}{9}=\frac{9}{99}\times\frac{66}{9}=\frac{6}{9}=0.\overline{6}$

What is the value of $\displaystyle \left ( 4.7\times13.26+4.7\times9.43+4.7\times77.31 \right )?$

maths calculations and mental strategies 4 mental additions and subtractions of decimals multiplication and division of decimals mental multiplication and division

  1. $470$

  2. $235$

  3. $705$

  4. $940$

Show answer
Correct Option: A
Explanation:

given that 

we have find the the value of the expression .

$4.7\times 13.26 +4.7 \times 9.43 +4.7 \times 77.31$

Taking $4.7$ common 

$= 4.7 \times [13.26 + 9.43 +77.31]$

$= 4.7 \times [100]$

$ = 470$

So option $A $ is correct